By a guaranteed error estimate for the gradient algorithm in Problem A we mean a number

.

By perturbations of problem A by means problem B

,

where is a non-decreasing -order-convex function on a partially set and .

Let be a guaranteed error estimate for the gradient algorithm in some unperturbed (perturbed) discrete optimization problem. As usual (see. [3]), we say that the gradient algorithm is stable if , where as .

Theorem. Let and be guaranteed error estimates for the gradient algorithm in Problems A and B, respectively. Then .

To prove Theorem, we need the following lemma.

Lemma. The gradient maximum and the global maximum of any -ordered-convex non-decreasing function on are connected by the following relations:

, (1)

where

– is the set of all maximal elements of the partially ordered set .

Proof of Lemma. By virtue of item of Theorem 4 [4], we have for

Together with the fact that

the last inequality yields

.

Therefore

,

Where

Then, by repeating the scheme of the proof of Theorem 4 [4], we obtain estimates (1). Lemma is proved.

Proof of Theorem. According to LemmaFinally we should note some rather confusing facts. Although given Jordan’s work on matrices and the fact that the Jordan normal form is named after him, the Gauss-Jordan pivoting elimination method for solving the matrix equation Ax= b is not. The Jordan of Gauss-Jordan is Wilhelm Jordan (1842 to 1899) who applied the method to finding squared errors to work on surveying. Jordan algebras are called after the German physicist and mathematician Pascual Jordan (1902 to 1980).

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• limit of a function as x approaches a fixed constant

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• limit of a function using the precise epsilon/delta definition of limit

• limit of a function using l’Hopital’s rule

,

,

where – is a non-decreasing -order-convex function on a partially set .

Let be an optimal solution of Problem A, and let be the point obtained by the following iterative procedure [4]:

which halts on the step if either or is the maximal element of the set (the set contains the zero , as we have stipulated). This point is called the gradient maximum os the function on the set [4].

By a guaranteed error estimate for the gradient algorithm in Problem A we mean a number

.

By perturbations of problem A by means problem B

,

where is a non-decreasing -order-convex function on a partially set and .

Let be a guaranteed error estimate for the gradient algorithm in some unperturbed (perturbed) discrete optimization problem. As usual (see. [3]), we say that the gradient algorithm is stable if , where as .

Theorem. Let and be guaranteed error estimates for the gradient algorithm in Problems A and B, respectively. Then .

To prove Theorem, we need the following lemma.

Lemma. The gradient maximum and the global maximum of any -ordered-convex non-decreasing function on are connected by the following relations:

, (1)

where

– is the set of all maximal elements of the partially ordered set .

Proof of Lemma. By virtue of item of Theorem 4 [4], we have for

Together with the fact that

the last inequality yields

.

Therefore

,

Where

]]>The centre-left may have lost in the battle of ideas in the 1980s, but we are winning now. And we have won a bigger battle today: the battle of values. The challenge we face has to be met by us together: one nation, one community.

When a young ]]>

H e n ( x ) = ( − 1 ) n e x 2 2 d n d x n e − x 2 2 = ( x − d d x ) n ⋅ 1 , {\displaystyle {\mathit {He}}_{n}(x)=(-1)^{n}e^{\frac {x^{2}}{2}}{\frac {d^{n}}{dx^{n}}}e^{-{\frac {x^{2}}{2}}}=\left(x-{\frac {d}{dx}}\right)^{n}\cdot 1,} {\displaystyle {\mathit {He}}_{n}(x)=(-1)^{n}e^{\frac {x^{2}}{2}}{\frac {d^{n}}{dx^{n}}}e^{-{\frac {x^{2}}{2}}}=\left(x-{\frac {d}{dx}}\right)^{n}\cdot 1,}

while the “physicists’ Hermite polynomials” are given by

H n ( x ) = ( − 1 ) n e x 2 d n d x n e − x 2 = ( 2 x − d d x ) n ⋅ 1. {\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}=\left(2x-{\frac {d}{dx}}\right)^{n}\cdot 1.} {\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}=\left(2x-{\frac {d}{dx}}\right)^{n}\cdot 1.}

These two definitions are not exactly identical; each is a rescaling of the other:

H n ( x ) = 2 n 2 H e n ( 2 x ) , H e n ( x ) = 2 − n 2 H n ( x 2 ) . {\displaystyle H_{n}(x)=2^{\frac {n}{2}}{\mathit {He}}_{n}\left({\sqrt {2}}\,x\right),\quad {\mathit {He}}_{n}(x)=2^{-{\frac {n}{2}}}H_{n}\left({\frac {x}{\sqrt {2}}}\right).} {\displaystyle H_{n}(x)=2^{\frac {n}{2}}{\mathit {He}}_{n}\left({\sqrt {2}}\,x\right),\quad {\mathit {He}}_{n}(x)=2^{-{\frac {n}{2}}}H_{n}\left({\frac {x}{\sqrt {2}}}\right).}

These are Hermite polynomial sequences of different variances; see the material on variances below.

The notation He and H is that used in the standard references.[5] The polynomials Hen are sometimes denoted by Hn, especially in probability theory, because

1 2 π e − x 2 2 {\displaystyle {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}} {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}

is the probability density function for the normal distribution with expected value 0 and standard deviation 1.

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